Adaptive Bayesian density estimation in sup-norm

Naulet, Zacharie

doi:10.3150/21-bej1387

Cited by 2 publications

(4 citation statements)

References 26 publications

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“…Proof. Using the same argument as above (33), we obtain the lower bound. Lemma 4 gives the upper bound.…”

Section: Then π[Tmentioning

confidence: 90%

“…It remains to determine the credibility level of the set C n . From Theorem 1 and Lemma 7, the posterior contracts towards f 0 and the fT * converges to f 0 on an asymptotically certain event E, both at a faster rate than σ n (see (33)). Therefore, an application of the triangular inequality gives…”

Section: Proofs For Confidence Bandsmentioning

confidence: 95%

“…While the use of a general theory based on prior mass and testing [17,18] made a relatively broad L 2 -theory possible [27,38], results for the supremum norm are typically more delicate, as uniform testing rates required in [17] appear to be slower [20]. Recent advances on this front include [6,23,34,33,40]. The first supremum norm posterior rates for tree methods, optimal up to a logarithmic factor, were obtained in [11] in regression models; we refer to [11] for more context and references on rates for tree-based methods.…”

Section: Introductionmentioning

confidence: 99%

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Optional Pólya trees: Posterior rates and uncertainty quantification

Castillo

Randrianarisoa

2022

Electron. J. Statist.

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We consider statistical inference in the density estimation model using a tree-based Bayesian approach, with Optional Pólya trees as prior distribution. We derive near-optimal convergence rates for corresponding posterior distributions with respect to the supremum norm. For broad classes of Hölder-smooth densities, we show that the method automatically adapts to the unknown Hölder regularity parameter. We consider the question of uncertainty quantification by providing mathematical guarantees for credible sets from the obtained posterior distributions, leading to near-optimal uncertainty quantification for the density function, as well as related functionals such as the cumulative distribution function. The results are illustrated through a brief simulation study.

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“…Proof. Using the same argument as above (33), we obtain the lower bound. Lemma 4 gives the upper bound.…”

Section: Then π[Tmentioning

confidence: 90%

Section: Proofs For Confidence Bandsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optional Pólya trees: Posterior rates and uncertainty quantification

Castillo

Randrianarisoa

2022

Electron. J. Statist.

View full text Add to dashboard Cite

show abstract

“…While the use of a general theory based on prior mass and testing [17,18] made a relatively broad L 2theory possible [27,38], results for the supremum norm are typically more delicate, as uniform testing rates required in [17] appear to be slower [20]. Recent advances on this front include [6,23,34,33,40]. The first supremum norm posterior rates for tree methods, optimal up to a logarithmic factor, were obtained in [11] in regression models; we refer to [11] for more context and references on rates for tree-based methods.…”

mentioning

confidence: 99%